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%I #11 Nov 25 2023 08:43:14
%S 6,10,14,15,21,22,26,30,33,34,35,36,38,39,42,46,51,55,57,58,60,62,65,
%T 66,69,70,74,77,78,82,84,85,86,87,90,91,93,94,95,100,102,105,106,110,
%U 111,114,115,118,119,120,122,123,126,129,130,132,133,134,138,140
%N Heinz numbers of non-Look-and-Say partitions. Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.
%C First differs from A130092 (non-Wilf partitions) in lacking 216.
%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%e The terms together with their prime indices begin:
%e 6: (2,1) 46: (9,1) 84: (4,2,1,1)
%e 10: (3,1) 51: (7,2) 85: (7,3)
%e 14: (4,1) 55: (5,3) 86: (14,1)
%e 15: (3,2) 57: (8,2) 87: (10,2)
%e 21: (4,2) 58: (10,1) 90: (3,2,2,1)
%e 22: (5,1) 60: (3,2,1,1) 91: (6,4)
%e 26: (6,1) 62: (11,1) 93: (11,2)
%e 30: (3,2,1) 65: (6,3) 94: (15,1)
%e 33: (5,2) 66: (5,2,1) 95: (8,3)
%e 34: (7,1) 69: (9,2) 100: (3,3,1,1)
%e 35: (4,3) 70: (4,3,1) 102: (7,2,1)
%e 36: (2,2,1,1) 74: (12,1) 105: (4,3,2)
%e 38: (8,1) 77: (5,4) 106: (16,1)
%e 39: (6,2) 78: (6,2,1) 110: (5,3,1)
%e 42: (4,2,1) 82: (13,1) 111: (12,2)
%e For example, the prime indices of 150 are {1,2,3,3}, with permutations and run-lengths (right):
%e (3,3,2,1) -> (2,1,1)
%e (3,3,1,2) -> (2,1,1)
%e (3,2,3,1) -> (1,1,1,1)
%e (3,2,1,3) -> (1,1,1,1)
%e (3,1,3,2) -> (1,1,1,1)
%e (3,1,2,3) -> (1,1,1,1)
%e (2,3,3,1) -> (1,2,1)
%e (2,3,1,3) -> (1,1,1,1)
%e (2,1,3,3) -> (1,1,2)
%e (1,3,3,2) -> (1,2,1)
%e (1,3,2,3) -> (1,1,1,1)
%e (1,2,3,3) -> (1,1,2)
%e Since none have all distinct run-lengths, 150 is in the sequence.
%t Select[Range[100],Select[Permutations[Join@@ ConstantArray@@@FactorInteger[#]],UnsameQ@@Length/@Split[#]&]=={}&]
%Y Wilf partitions are counted by A098859, ranked by A130091.
%Y Non-Wilf partitions are counted by A336866, ranked by A130092.
%Y A variant for runs is A351201, counted by A351203 (complement A351204).
%Y These partitions are counted by A351293.
%Y The complement is A351294, counted by A239455.
%Y A032020 = number of binary expansions with distinct run-lengths.
%Y A044813 = numbers whose binary expansion has all distinct run-lengths.
%Y A056239 = sum of prime indices, row sums of A112798.
%Y A165413 = number of distinct run-lengths in binary expansion.
%Y A181819 = Heinz number of prime signature (prime shadow).
%Y A182850/A323014 = frequency depth, counted by A225485/A325280.
%Y A297770 = number of distinct runs in binary expansion.
%Y A320922 ranks graphical partitions, complement A339618, counted by A000569.
%Y A329739 = compositions with all distinct run-lengths, for all runs A351013.
%Y A329747 = runs-resistance, counted by A329746.
%Y A333489 ranks anti-runs, complement A348612.
%Y A351017 = binary words with all distinct run-lengths, for all runs A351016.
%Y Cf. A000961, A001221, A001222, A175413, A182857, A304660, A320924, A328592, A351202, A351290.
%K nonn
%O 1,1
%A _Gus Wiseman_, Feb 16 2022